You wouldn't imagine Duo!
Is this a literal question or does it have a general connotation as it does in English?
I've not heard it used as an ice breaker, but if you were to say it to a stranger (who also speaks hindi) they would get you're asking for their life story.
I wonder if this question can have the colloquial, maybe somewhat vulgar meaning, if said with impatience: What is wrong with you?
No, कहानी is too formal to be used in that sense.
The closest to "What is the matter with you?" would be क्या हो गया तुझे ? ("What happened to you?" )